The canonical homomorphisms of topological -spaces are -equivariant continuous functions, and the canonical choice of homotopies between these are -equivariant continuous homotopies (for trivial -action on the interval). A -equivariant version of the Whitehead theorem says that on G-CW complexes these -equivariant homotopy equivalences are equivalently those maps that induce weak homotopy equivalences on all fixed point spaces for all subgroups of (compact subgroups, if is allowed to be a Lie group). By Elmendorf's theorem, this, in turn, is equivalent to the (∞,1)-presheaves over the orbit category of . See below at In topological spaces – Homotopy theory.
(Beware that -homotopy theory is crucially different from (namely “finer” and “more geometric” than) the homotopy theory of the ∞-actions of the underlying homotopy type ∞-group of , and this is so even when is a discrete group, see below).
The union of -equivariant homotopy theories as is allowed to vary is global equivariant homotopy theory.
The direct stabilization of equivariant homotopy theory is the theory of spectra with G-action. More generally there is a concept of G-spectra and they are the subject of equivariant stable homotopy theory.
|Borel equivariant cohomology||general (Bredon) equivariant cohomology||non-equivariant cohomology with homotopy fixed point coefficients|
|trivial action on coefficients||trivial action on domain space|
Let be a discrete group.
(e.g. May 96, p. 15)
(models for -equivariant spaces)
Consider the following three homotopical categories that model -spaces:
for the full subcategory of G-CW-complexes, regarded as equipped with the structure of a category with weak equivalences by taking the weak equivalences to be the - homotopy equivalences according to def. 2.
for all of equipped with weak equivalences given by those morphisms that induce on for all subgroups weak homotopy equivalences on the -fixed point spaces, in the standard Quillen model structure on topological spaces (i.e. inducing isomorphism on homotopy groups).
This is stated as (May 96, theorem VI.6.3).
At topological ∞-groupoid it is discussed that the category Top of topological spaces may be understood as the localization of an (∞,1)-category of (∞,1)-sheaves on , at the collection of morphisms of the form with the real line.
The analogous statement is true for -spaces: the equivariant homotopy category is the homotopy localization of the category of -stacks on .
More in detail: let be the site whose objects are -spaces that admit -equivariant open covers, morphisms are -equivariant maps and morphism is in the coverage if it admits a -equivariant splitting over such -equivariant open covers.
for the left Bousfield localization at thecollection of morphisms .
Then the homotopy category of is the equivariant homotopy category described above
This is (Morel-Voevodsky 03, example 3, p. 50).
The above constructions may be unified to apply “for all groups at once”, this is the content of global equivariant homotopy theory.
Let be a cofibrantly generated model category with generating cofibrations and generating acyclic cofibrations .
There is a cofibrantly generated model category
and the generating acyclic cofibrations to be
and the generating acyclic cofibrations to be
This defines a cofibrantly generated model category if has a cellular fixed point functor (see…).
(generalized Elmendorf’s theorem)
There is a Quillen adjunction
and a Quillen equivalence
This is proposition 3.1.5 in Guillou.
For instance, in motivic homotopy theory one considers cohomology in a homotopy localization of the ∞-stack (∞,1)-topos on the Nisnevich site, presented by . Its -equivariant version as above should be the right context for the Bredon -equivariant cohomology refinement of such cohomology theories, such as motivic cohomology.
This is example 4.5 in Guillou.
For a discrete group (geometrically discrete) the homotopy theory of G-spaces which enters Elmendorf's theorem is different (finer) than the standard homotopy theory of -∞-actions, which is presented by the Borel model structure (see there for more, and see (Guillou)).
|cohesive (∞,1)-topos||its (∞,1)-site||base (∞,1)-topos||its (∞,1)-site|
|global equivariant homotopy theory||global equivariant indexing category||∞Grpd||point|
|… sliced over terminal orbispace:||orbispaces||global orbit category|
|… sliced over :||-equivariant homotopy theory of G-spaces||-orbit category|
A standard text is
For a brief modern surves see also the first three sections of
(with an eye towards application to the Arf-Kervaire invariant problem)
The generalization of the homotopy theory of -spaces and of Elmendorf’s theorem to that of -objects in more general model categories is in
and further discussed in
Discussion in the context of global equivariant homotopy theory is in
Discussion via homotopy type theory is in